We give an example of a compact set K ⊂ [0, 1] such that the space ℇ(K) of Whitney functions is isomorphic to the space s of rapidly decreasing sequences, and hence there exists a linear continuous extension operator L:ℇ(K)→C∞[0,1]. At the same time, Markov’s inequality is not satisfied for certain polynomials on K.

We prove that for each countably infinite, regular space X such that Cp(X) is a Zσ-space, the topology of Cp(X) is determined by the class F0(Cp(X)) of spaces embeddable onto closed subsets of Cp(X). We show that Cp(X), whenever Borel, is of an exact multiplicative class; it is homeomorphic to the absorbing set Ωα for the multiplicative Borel class Mα if F0(Cp(X))=Mα. For each ordinal α ≥ 2, we provide an example Xα such that Cp(Xα) is homeomorphic to Ωα.

We construct two examples of infinite spaces X such that there is no continuous linear surjection from the space of continuous functions cp(X) onto cp(X) × ℝ.Inparticular,cp(X)isnotlinearlyhomeomorphictocp(X)×ℝ. One of these examples is compact. This answers some questions of Arkhangel’skiĭ.

We show that Whitney?s approximation theorem holds in a general setting including spaces of (ultra)differentiable functions and ultradistributions. This is used to obtain real analytic modifications for differentiable functions including optimal estimates. Finally, a surjectivity criterion for continuous linear operators between Fréchet sheaves is deduced, which can be applied to the boundary value problem for holomorphic functions and to convolution operators in spaces of ultradifferentiable functions...

Let X be an infinite, locally connected, locally compact separable metrizable space. The space C(X) of real-valued continuous functions defined on X with the compact-open topology is a separable Fréchet space, so it is homeomorphic to the psuedo-interior s = (−1, 1)ℕ of the Hilbert cube Q = [−1, 1]ℕ. In this paper, generalizing the Sakai-Uehara’s result to the non-compact case, we construct a natural compactification C¯
(X) of C(X) such that the pair (C¯
(X), C(X)) is homeomorphic to (Q, s). In case...

We characterize composition operators on spaces of real analytic functions which are open onto their images. We give an example of a semiproper map φ such that the associated composition operator is not open onto its image.